finite elements in electromagnetics 1. introduction
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Finite Elements in Electromagnetics 1. Introduction. Oszkár Bíró IGTE, TU Graz Kopernikusgasse 24, Graz, Austria email: [email protected]. Overview. Maxwell‘s equations Boundary value problems for potentials Nodal finite elements Edge finite elements. Maxwell‘s equations. - PowerPoint PPT PresentationTRANSCRIPT
Finite Elements in Electromagnetics1. Introduction
Oszkár BíróIGTE, TU Graz
Kopernikusgasse 24, Graz, Austriaemail: [email protected]
Overview
• Maxwell‘s equations• Boundary value problems for potentials• Nodal finite elements• Edge finite elements
Maxwell‘s equations
DB
BE
DJH
divdiv
tcurl
tcurl
0
EDJEEJBHHB ;,;,
Potentials
tVgrad
t
curl
AE
AB
• Continuous functions• Satisfy second order differential equations• Neumann and Dirichlet boundary conditions
E.g. magnetic vector and electric scalar potential (A,V formulation):
Differential equations
:0DJH t
curl
0AAA
2
2
2
2
)(tVgrad
ttVgrad
tcurlcurl
:0)( t
div DJ
0)( 2
2
2
2
tVgrad
ttVgrad
tdiv AA
E
H
in a closed domain
Dirichlet boundary conditions
AnnBnnAnE curltVgrad
t
,
Prescription of tangential E (and normal B) on E:
0
,VV 0anA
n is the outer unit normal at the boundary
E
H
nEB
Neumann boundary conditions Prescription of tangential H (and normal J+JD) on H:
j)()(
,
2
2
2
2
nAnAKnA
tVgrad
ttVgrad
t
curl
nAnAnDJ
nAnH
)()()(
,
2
2
2
2
tVgrad
ttVgrad
tt
curl
E
H n
H
J+JD
General boundary value problem
in 2
2
212 ftuL
tuLuL tt
Differential equation:
Boundary conditions:DDuL on 0
NNtNtN gtuL
tuLuL
on 2
2
21
Dirichlet BC
Neumann BC
D
N
ND
Nonhomogeneous Dirichlet boundary conditions
uuu D DD uuL on 0
ithfunction wunknown new :uDDDD uuLu on that soarbitrary : 0
2
2
2122
2
212 tuL
tuLuLf
tuL
tuLuL D
tD
tDtt
2
2
212
2
21 tuL
tuLuLg
tuL
tuLuL D
NtD
NtDNNtNtN
DDuL on 0
Formulation as an operator equation (1)
Characteristic function of a domain
. if ,0, if ,1
)(PP
P wwdw
,,
wwdw
,,
Dirac function of a surface
gradn
Scalar product for ordinary functions:
3
, uvdvu
Formulation as an operator equation (2)
uLuLAu NN 2
gftuC
tuBAu
N
2
2
Define the operators A, B and C as
(with the definition set})on 0:{ DDABC uLuD
Equivalent operator equation:
uLuLCu Ntt N 22 uLuLBu Ntt N 11
Formulation as an operator equation (3)Properties of the operators:
Symmetry: ,,, AwuwAu ,,, BwuwBu .,,,, ABCDwuCwuwCu
Positive property: ABCDuuAu ,0,
Operators of the A,V formulation (1)
V
uA
000)( ncurlcurlcurl
A H
gradgraddivdivgrad
BHHnn
)()(
}on 0,:{ EA VV
D
0nAA
gradgraddivdivgrad
CHHnn
)()(
A,V formulation: symmetry of A
w
w
u
u
VVA
AA,
3
)( dcurlcurlcurl wuu HAnAA
H
dcurldcurlcurl uwuw )()( nAAAA
dcurldcurlcurl uwuw nAAAA )(
H
dcurl uw )( nAA
Ew
E
dcurldcurlcurl uwuw
on since ,0
)(
0nA
AnAAA
dcurlcurl uw AA
A,V formulation: positive property of A
dcurlcurl
VVA AA
AA,
02
dcurlA
A,V formulation: symmetry of B and C
w
w
u
u
VVB
AA,
3 )())((
)(d
VgradVgradVdivgradV
wuuuu
wuu
HAnA
AA
dgradVdivVgradV uuwuuw )]([)( AAA
H
dgradVV uuw nA )(
dgradVgradVgradV uuwuuw )()( AAA
H
dgradVVdgradVV uuwuuw nAnA )()(
dgradVgradVgradVgradV uwuwuwuw AAAA
Ew
E
V
uuw dgradVV
on 0 since ,0
)( nA
dgradVgradVgradVgradV uwuwuwuw AAAA
Weak form of the operator equation
ABCDwwgfwtuC
tuBAu
N
,,,2
2
Galerkin’s method:discrete counterpart of the weak form
n
kkk
n ftuutu1
)( )()(),( rr ABCk Df
ABCk Dkf in set entirean forming functions basis: ,...2,1,
,,,2
)(2)()(
ii
nnn fgff
tuC
tuBAu
N
ni ..., 2, 1,
Set of ordinary differential equations
Galerkin equations buCuBuA
kiikikikik aAfffAfaaA ,,,
[A] is a symmetric positive matrix kiikikikik bBfffBfbbB ,,, kiikikikik cCfffCfccC ,,,
[B] and [C] are symmetric matrices iii fgfbbb
N,,
Finite element discretization
Nodal finite elements (1)
12
3
4
5
6
7
8
9
10
11
12
1314
15
16
17
18
1920
nodes.other allin 0, nodein 1
)(i
N i r i = 1, 2, ..., nn
Shape functions:
Nodal finite elements (2)Shape functions
Corner node Midside node
Nodal finite elements (3)
Basis functions for scalar quantities (e.g. V): Shape functions
Number of nodes: nn, number of nodes on D: nDn,Dnn nnn nodes on D: n+1, n+2, ..., nn
n
kkk
n NtVVtV1
)( )()(),( rr
Nodal finite elements (4)Linear independence of nodal shape functions
11
nn
iiN
Taking the gradient:
01
nn
iiNgrad
The number of linearly independent gradients of the shape functions is nn-1 (tree edges)
Edge finite elements (1)
12
3
4
56
7
8
9
10
11
12
13
14
15
16
1718
19
20
22
23
24
25
26
27
28
29
30 31
32
33 34
35
36
21
Edge basis functions:
. if , 0, if , 1
)(jiji
djEdgei lrN i = 1, 2, ..., ne
Edge finite elements (2)Basis functions
Side edge Across edge
Edge finite elements (3)
Basis functions for vector intensities (e.g. A): Edge basis functions
Number of edges: ne, number of edges on D: nDe,Dee nnn edges on D: n+1, n+2, ..., ne
n
kkk
n tat1
)( )()(),( rNArA
Edge finite elements (4)Linear independence of edge basis functions
Taking the curl:
The number of linearly independent curls of the edge basis functions is ne-(nn-1) (co-tree edges)
;1
en
kkiki cgradN N 0
1
2
en
kikc i=1,2,...,nn-1.
,1
0N
en
kkikcurlc i=1,2,...,nn-1.